Linear Function

Definition

A linear function is a mathematical function where the relationship between the independent and dependent variables can be represented by a straight line. This type of function is characterized by a constant rate of change, known as the slope, and an initial value, known as the y-intercept.

5 Must Know Facts For Your Next Test

1. Linear functions have a constant rate of change, which means that for every unit change in the independent variable, the dependent variable changes by the same amount.
2. The graph of a linear function is always a straight line, which can be either increasing, decreasing, or horizontal.
3. The slope of a linear function can be positive, negative, or zero, indicating the direction and steepness of the line.
4. Linear functions can be used to model a wide range of real-world situations, such as the relationship between distance and time, or the cost of a product and the quantity purchased.
5. The y-intercept of a linear function represents the value of the dependent variable when the independent variable is zero.

Review Questions

• Explain the relationship between the slope and the rate of change in a linear function.
  • The slope of a linear function represents the constant rate of change between the independent and dependent variables. The slope indicates the steepness and direction of the line, where a positive slope represents an increasing relationship, a negative slope represents a decreasing relationship, and a slope of zero represents a horizontal line. The slope can be calculated as the change in the dependent variable divided by the change in the independent variable, and this ratio remains constant throughout the function.
• Describe how the y-intercept of a linear function relates to the initial value of the function.
  • The y-intercept of a linear function represents the value of the dependent variable when the independent variable is zero. This initial value is the point where the line crosses the y-axis and provides important information about the starting point or baseline of the function. The y-intercept, along with the slope, determines the equation of the linear function, which can be used to make predictions and analyze the relationship between the variables.
• Analyze how the graph of a linear function can be used to identify and interpret the key characteristics of the function.
  • The graph of a linear function provides a visual representation of the relationship between the independent and dependent variables. From the graph, you can identify the slope of the line, which indicates the rate of change, and the y-intercept, which represents the initial value of the function. The direction and steepness of the line can be used to determine whether the function is increasing, decreasing, or horizontal. Additionally, the graph can be used to make predictions about the values of the dependent variable for given values of the independent variable, as well as to identify any points of intersection or special points on the line.

Related terms

Slope: The slope of a linear function represents the constant rate of change between the independent and dependent variables. It indicates the steepness and direction of the line.

Y-Intercept: The y-intercept of a linear function is the point where the line crosses the y-axis, representing the initial value of the function.

Equation of a Line: The equation of a linear function is typically written in the form $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.